(a)
\(\mu ^A\)  is the unique measure in \(\mathfrak {M}^+\)  satisfying the symmetry relation
\(\kappa _\alpha (\mu ^A,\sigma )=\kappa _\alpha (\sigma ^A,\mu )\text{ \ for all $\sigma \in \mathcal {E}_\alpha ^+$},\) 
where \(\sigma ^A\)  denotes the only measure in \(\mathcal {E}^{\prime }_A\)  with \(\kappa _\alpha \sigma ^A=\kappa _\alpha \sigma \)  n.e. on \(A\) .For any \(\sigma \in \mathcal {E}^+_\alpha \)  and any \(A\subset \mathbb {R}^n\) , the measure \(\sigma ^A\in \mathcal {E}^{\prime }_A\)  having the property \(\kappa _\alpha \sigma ^A=\kappa _\alpha \sigma \)  n.e. on \(A\) , exists and is unique. It is in fact the orthogonal projection of \(\sigma \)  in the pre-Hilbert space \(\mathcal {E}_\alpha \)  onto the convex, strongly complete cone \(\mathcal {E}^{\prime }_A\) ; that is (compare with (REF )),
\(\Vert \sigma -\sigma ^A\Vert =\min _{\nu \in \mathcal {E}^{\prime }_A}\,\Vert \sigma -\nu \Vert .\) 
Alternatively, \(\sigma ^A\)  is uniquely characterized within \(\mathfrak {M}^+\)  by extremal property (REF ) with \(\mu :=\sigma \) .

(b)
\(\mu ^A\)  is the unique measure in \(\mathfrak {M}^+\)  satisfying either of the two limit relations
\(&\mu _j^A\rightarrow \mu ^A\text{ \ vaguely in $\mathfrak {M}^+$ as $j\rightarrow \infty $},\\&\kappa _\alpha \mu _j^A\uparrow \kappa _\alpha \mu ^A\text{ \ pointwise on $\mathbb {R}^n$ as $j\rightarrow \infty $},\) 

where \((\mu _j)\)  is an arbitrary sequence of \(\mu _j\in \mathcal {E}^+_\alpha \)  having the propertySuch \(\mu _j\in \mathcal {E}^+_\alpha \) , \(j\in \mathbb {N}\) , do exist; they can be defined, for instance, by means of the formula
\(\kappa _\alpha \mu _j:=\min \,\bigl \lbrace \kappa _\alpha \mu ,\,j\kappa _\alpha \lambda \bigr \rbrace ,\) 
\(\lambda \in \mathcal {E}^+_\alpha \)  being fixed (see e.g. [1]} or [2]}). Here we have used the fact that for any \(\mu _1,\mu _2\in \mathfrak {M}^+\) , there is \(\mu _0\in \mathfrak {M}^+\)  such that \(\kappa _\alpha \mu _0:=\min \,\lbrace \kappa _\alpha \mu _1,\,\kappa _\alpha \mu _2\rbrace \)  [1]}.

\(\kappa _\alpha \mu _j\uparrow \kappa _\alpha \mu \text{ \ pointwise on $\mathbb {R}^n$ as $j\rightarrow \infty $},\) 
whereas \(\mu _j^A\)  denotes the only measure in \(\mathcal {E}^{\prime }_A\)  with \(\kappa _\alpha \mu _j^A=\kappa _\alpha \mu _j\)  n.e. on \(A\) . (Regarding the existence and uniqueness of this \(\mu _j^A\) , see footnote REF ).


